Hierarchic models for the modal analyses of ceramic–metal functional gradient (FG) plates are introduced. This study was motivated by the fact that the free vibration analyses of those structures are usually conducted by the first- or higher-order shear deformation theories (SDT). Meanwhile, a hierarchic model does not only include these classical theories, but their model level varies from the first order to the 3D full elasticity. In fact, in the hierarchic models, the part of displacement field through the thickness is assumed a priori using polynomials with the desired highest order. Hence, they can be numerically implemented using 2D finite element or mesh-free method, by discretizing only the mid-surface of 3D structures. The global mass and global stiffness matrices of FG plates are computed by adopting the Gaussian quadrature rules for the mid-surface integral and the trapezoidal rule for the thickness-wise integral. The proposed hierarchic models are numerically illustrated and their characteristics are investigated. In addition, the modal characteristics of FG plates are parametrically examined to the key factors of gradient layer. The hierarchic models approach the same limit and show a sequence of model accuracy. Meanwhile, the modal responses of metal-ceramic FG plate structures are dependent of the volume fraction pattern and the relative thickness ratio of gradient layer.

介绍了用于陶瓷-金属功能梯度（FG）板模态分析的层次模型。这项研究的动机是，这些结构的自由振动分析通常是由一阶或更高阶的剪切变形理论（SDT）进行的。同时，层次模型不仅包含这些经典理论，而且其模型级别也从一阶到3D全弹性而变化。实际上，在分层模型中，使用具有所需最高阶的多项式先验地假设了整个厚度范围内的位移场。因此，通过仅离散化3D结构的中表面，可以使用2D有限元或无网格方法在数值上实现它们。FG板的整体质量和整体刚度矩阵是通过对中表面积分采用高斯正交规则而对厚度方向积分采用梯形规则来计算的。对提出的分层模型进行了数值说明，并研究了它们的特性。另外，对梯度板的关键因素进行了参数化检查。层次模型接近相同的限制，并显示出一系列的模型准确性。同时，金属陶瓷FG板结构的模态响应取决于体积分数模式和梯度层的相对厚度比。另外，对梯度板的关键因素进行了参数化检查。层次模型接近相同的限制，并显示出一系列的模型准确性。同时，金属陶瓷FG板结构的模态响应取决于体积分数模式和梯度层的相对厚度比。另外，对梯度板的关键因素进行了参数化检查。层次模型接近相同的限制，并显示出一系列的模型准确性。同时，金属陶瓷FG板结构的模态响应取决于体积分数模式和梯度层的相对厚度比。